T-test with imperfectly-measured data Weird question here that we never covered in class
Say you’re running a 2-sample t test to see if feeding children candy stunts their growth. Your null hypothesis is the candy-fed fiends are at least as tall as the veggie kids; the alternate is that the candy fiends are shorter
Simple enough, right?
But what if the woman taking their heights isn’t so accurate?
Say you had her measure a yardstick a hundred times. You determined there was a normal-ish distribution centered around a yard, with standard deviation of 10.8 inches (30%). Her measurements are bad
The 30% standard deviation holds for her measurements of everything and every length/height (let’s say)
{real-life version might be measuring tiny things, or fast-moving things, or some outcomes in the medical realm}
Now what kind of math helps here?
Surely it cannot be just a regular t test.
The measurement variation must cause all kinds of problems
The furthest I’ve thought is something with a convolution, but that’s no real help
Thanks!
 A: 
Surely it cannot be just a regular t test.

It sure can.  As you've described it, the heights, $H$ of children in each condition could reasonably be modelled as normal with some mean $\mu$ and variance $\sigma^2$.
$$ H \sim \mathcal{N}(\mu, \sigma^2) \>. $$
Now, the data are corrupted via measurement error.  The observed height, $\hat{H}$ of each child is further normal with variance $10.8^2$
$$ \hat{H} \sim \mathcal{N}(H, 10.8^2) \>.$$
This means that $\hat{H}$ is normal too (worth while to ask yourself why), and you can just use the t test.  The measurement error is combined with the variation in child heights.  So long as the measurer is unbiased (e.g. she is really short can can't reach the heads of really tall children) then nothing but statistical power and precision is lost in using the t-test.
A: This is just one of the many assumptions that aren't made explicit when presenting analyses. When we "slap an error term" on the probability model for the response, we don't usually believe in a deep, fundamental level of randomness. Rather we consider the replicability of the design and a range of effects that might be typical of a particular observation if the study were repeated. Intrinsic to this includes the measurement error, the selection of the particular student, their posture, the time of day at which the measurement was performed, etc. etc. etc. The only thing (in a T-test) that we "account for" so to speak is the actual group assignment. Everything else we are confident to call "random" if we believe in the homogeneity and independence of the residuals - "random" has a particular meaning here.
When certain variables are measured, and contribute to the error, we can further block or adjust for their effects in an ANOVA or linear model. For measurement error, we might go farther and look at measurer effects (such as who does the particular measuring) and consider adjusting for their influence by way of a random effect. This is particularly important in, say, breast cancer imaging studies where radiologists interpret scans - and they don't always agree. In that case we need to adjust for an "interpreter" effect.
